Integrability of the Zakharov-Shabat Systems by Quadrature

Abstract

We consider the general two-dimensional Zakharov-Shabat systems, which appear in application of the inverse scattering transform (IST) to an important class of nonlinear partial differential equations (PDEs) called integrable systems. We study their integrability in the meaning of differential Galois theory, i.e., their solvability by quadrature. It becomes a key for obtaining analytical expressions for solutions to the PDEs by using the IST. For a wide class of potentials, we prove that they are integrable in that meaning if and only if the potentials are reflectionless. It is well known that for such potentials particular solutions called n-solitons in the original PDEs are yielded by the IST.